According to the source novel it was done (largely) as a tax dodge. Having multiple parents means that each parent can claim a tax break to support their child. Eight parents means eight allowances but with only one additional mouth to feed.
“So many parents for only one child,” Lopez said, slowly unwrapping another lozenge. The Martians had lots of space for traditional families.
“The tax break for eight adults only having one child allowed them to own twenty-two acres of decent farmland. There are over thirty billion people on Earth. Twenty-two acres is a national park,” Holden said. “Also, the DNA mix is legit. They aren’t parents in name only.”
“How did they decide who carried you?”
“Mother Elise had the widest hips.”
Leviathan Wakes - Chapter Eleven: Holden
It would appear that his parents weren't just linked for legal purposes, they were also engaged in a complex and shifting polyamorous relationship.
They’d talked about how to break the news to the crew but hadn’t come
to any consensus. Holden hated to hide anything. Keeping it secret
made it seem dirty or shameful. His parents had raised him to believe
that sex was something you did in private not because it was
embarrassing, but because it was intimate. With five fathers and three
mothers, the sleeping arrangements were always complex at his house,
but the discussions about who was bedding with whom were never hidden
from him. It left him with a strong aversion to hiding his own
Leviathan Wakes - Chapter Forty-Seven: Holden